Interpreting Equivariant Representations

Latent representations are used extensively for downstream tasks, such as visualization, interpolation or feature extraction of deep learning models. Invariant and equivariant neural networks are powerful and well-established models for enforcing inductive biases. In this paper, we demonstrate that the inductive bias imposed on the by an equivariant model must also be taken into account when using latent representations. We show how not accounting for the inductive biases leads to decreased performance on downstream tasks, and vice versa, how accounting for inductive biases can be done effectively by using an invariant projection of the latent representations. We propose principles for how to choose such a projection, and show the impact of using these principles in two common examples: First, we study a permutation equivariant variational auto-encoder trained for molecule graph generation; here we show that invariant projections can be designed that incur no loss of information in the resulting invariant representation. Next, we study a rotation-equivariant representation used for image classification. Here, we illustrate how random invariant projections can be used to obtain an invariant representation with a high degree of retained information. In both cases, the analysis of invariant latent representations proves superior to their equivariant counterparts. Finally, we illustrate that the phenomena documented here for equivariant neural networks have counterparts in standard neural networks where invariance is encouraged via augmentation. Thus, while these ambiguities may be known by experienced developers of equivariant models, we make both the knowledge as well as effective tools to handle the ambiguities available to the broader community.